This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A386760 #24 Aug 28 2025 10:23:44
%S A386760 5,6,10,11,15,16,20,24,25,29,30,34,35,39,44,48,49,53,54,58,59,63,67,
%T A386760 68,72,73,77,78,82,83,87,91,92,96,97,101,102,106,111,115,116,120,121,
%U A386760 125,126,130,134,135,139,140,144,145,149,150,154,158,159,163,164,168
%N A386760 Numbers k such that the number of decimal digits of the Lucas number L(k) is greater than the number of decimal digits of the Fibonacci number F(k).
%C A386760 The difference in the number of decimal digits, A055642(L(k))-A055642(F(k)) = A060384(k)-A386758(k) is either zero or one. In fact, this difference is ceiling(beta-{k*alpha}), with alpha and beta as defined in the Formula section. This implies that, asymptotically, a fraction of beta=0.349485... of the Lucas numbers has one more decimal digit than the corresponding Fibonacci number. This gives the asymptotic behavior of the sequence as a(n)~n/beta. Conjecture: abs(a(n)-n/beta)<c, for some constant c.
%H A386760 Hans J. H. Tuenter, <a href="/A386760/b386760.txt">Table of n, a(n) for n = 1..1000</a>
%F A386760 The sequence consists of the integers k>=2, for which {k*alpha}<beta, where alpha=log_10(phi), beta=log_10(5)/2, {x}=x-floor(x), denotes the fractional part of x, log_10(phi) = A097348, and phi = (1+sqrt(5))/2 = A001622.
%e A386760 5 is a term since F(5)=5 has length 1 decimal digit, but L(5)=11 has length 2 decimal digits which is greater.
%t A386760 Select[Range[168],IntegerLength[LucasL[#]]>IntegerLength[Fibonacci[#]]&] (* _James C. McMahon_, Aug 28 2025 *)
%Y A386760 Cf. A000032, A000045, A001622, A055642, A060384, A097348, A386758.
%K A386760 base,nonn,easy,new
%O A386760 1,1
%A A386760 _Hans J. H. Tuenter_, Aug 13 2025